Pome

Proposition 16.3.29B9461C380E5C

Proposition 16.3.29.B9461C380E5C Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Assume that

8997CB842099 16.3.30
\begin{equation} \varphi =P_{\theta }[\varphi ],\quad \psi =P_{\theta }[\psi ],\quad \varphi \land \psi \not\equiv -\infty . \end{equation}

Then

7E981960ACBC 16.3.31
\begin{equation} P_{\theta }[\varphi \land \psi ]=\varphi \land \psi . \end{equation}

Proof. Observe that by proposition 16.4.2 3,

\[ P_{\theta }[\varphi \land \psi ]\leq P_{\theta }[\varphi ]=\varphi ,\quad P_{\theta }[\varphi \land \psi ]\leq P_{\theta }[\psi ]=\psi . \]

So the $\leq $ direction in (16.3.31) holds. The reverse direction is trivial. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub